Cryptograpy

1
Cryptograpy

• By Roya Furmuly

9
O
D
I
C
S
L
1
7
W
3
F
P
2
H
2
What Is It?

• Enables two people (Alice and Bob) to
  communicate over an insecure channel in such a
  way so that an opponent (Oscar) cannot understand
  what is being said.

3
How Does It Work?

• Alice encrypts the information (Plaintext), using
  a predetermined key, then sends the result
  (Ciphertext) to Bob.
• Oscar cannot determine the plaintext because he
  doesnt know the key.
• Bob, who knows the encryption key, decrypts the
  ciphertext and reconstructs the plaintext.

4
Formal Definition

• A Cryptosystem is a five-tuple (P,C,K,E,D )
• P finite set of plaintexts
• C finite set of ciphertexts
• K finite set of keys (keyspace)
• For each K K eK E and a
  corresponding dK D. Each eKP C and dKC
  P are functions such that dK(eK(x))x x
  P.

5
Observations

• The encryption function eK must be injective to
  avoid ambiguity.
• i.e. if y eK(x1) eK(x2) where x1 not
  equal x2
• Bob doesnt know whether y x1 or y x2
• If P C , then the encryption function is a
  permutation.

6
Protocol

• Choose random key K in K (when Oscar not present
  or through a secure channel).
• Alice
• Message xx1x2...xn where i in (1,n),
  xi in P
• encrypts each xi using encryption rule yi
  eK(xi) yy1y2yn
• Bob uses decryption function dK(yi)xi
  
  xx1x2...xn

7
Diagram
Oscar
Oscar
x
y
x
Alice
encrypter
decrypter
Bob
K
key source
8
What makes a Cryptosystem practical?

• 1. Encryption and Decryption functions should be
  efficiently computable.
• 2. Upon seeing ciphertext y, the opponent should
  be unable to determine the key K used
  (security).

9
Shift Cipher

• Let P C K Z26.
• eK(x)xK mod 26
• and
• dK(y)y-K mod 26 (x,y
  in Z26)
• cool fact for K3, cryptosystem is
  called the Caesar Cipher.

10
Shift Cipher (contd)

• We encrypt English text, by the following
  correspondence
• A 0, B 1, , Z 25,
• A B C D E F G H I J K L M N O P Q R S T U V W
• 0 1 2 3 4 5 6 7 8 9 101112 13 14
  15161718192021 22
• X Y Z
• 23 24 25

11
Lets Encrypt!

• Let the key be K7, encrypt UCLA BRUINS
• convert letters to integers using chart
• 20 2 11 0 1 17 20 8 13 18
• add 7 to each value, reduce mod 26
• 1 9 18 7 8 24 1 15 20 25
• convert to sequence of integers
• BJSHIYBPUZ

12
Lets Decrypt!

• BJSHIYBPUZ
• convert letters to integers
• 1 9 18 7 8 24 1 15 20 25
• subtract 7, reduce mod 26
• 20 2 11 0 1 17 20 8 13 18
• convert to letters
• UCLA BRUINS

13
Shift Cipher, any Good?

• Nope! Fails security property.
• Keyspace is very small, only 25 possible keys.
• Can easily be deciphered by an exhaustive key
  search.
• Try K125, until get a text that makes sense.

14
Vigenere Cipher

• Let mgt0 be fixed. Let P C K (Z26)m
• For a key K(k1,k2,km) define
• eK(x1,x2,,xm)(x1k1, x2k2,,xmkm)
• and
• dK(y1,y2,,ym)(y1-k1, y2-k2,,ym-km)
• all operations done in Z26

15
Lets Encrypt!

• Let keyhot(7,14,19), encrypt SUMMER IS
  HERE
• convert to integers add the keyword mod 26
• 18 20 12 12 4 17 8 18 7 4 18 4
• 7 14 19 7 14 19 7 14 19 7 14 19
• --------------------------------------------------
  --
• 25 8 5 19 18 10 15 6 0 11 6 23
• ZIFTSKPGALGX

16
Lets Decrypt!

• ZIFTSKPGALGX
• convert to integers and subtract the keyword
  hot(7,14,19) mod 26
• 25 8 5 19 18 10 15 6 0 11 6
  23
• 7 14 19 7 14 19 7 14 19 7 14
  19
• --------------------------------------------------
  ------
• 18 20 12 12 4 17 8 18 7 4 18
  4
• SUMMER IS HERE

17
Vigenere Cipher, any Good?

• Better than Shift Cipher
• Possible number of keys of length m is
• (26)m
• Say m5, then keyspace size is
• (26)5 approx 1.1x107
• So, exhaustive key search not feasible by hand
  (but OK by computer).

18
Other Cryptosystems

• Data Encryption Standard (DES)
• Based on permutaion of 64 bits at a time.
• RSA
• Based on difficulty of factoring large
  integers into primes.
• Enigma
• Machine with rotors that shifted letters in
  complicated manner.

19
Summary

• Cryptography allows us to communicate through
  insecure channels.
• Shift Cipherinsecure (small keyspace)
• Vigenere Cipherless insecure
• Complicated Cryptosystems
• DES, RSA, ENIGMA

20
WKH HQG